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Your data matches 360 different statistics following compositions of up to 3 maps.
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Matching statistic: St000173
St000173: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 1
[[2,2]]
=> 1
[[1],[2]]
=> 0
[[1,3]]
=> 1
[[2,3]]
=> 2
[[3,3]]
=> 1
[[1],[3]]
=> 1
[[2],[3]]
=> 2
[[1,1,2]]
=> 1
[[1,2,2]]
=> 1
[[2,2,2]]
=> 1
[[1,1],[2]]
=> 0
[[1,2],[2]]
=> 1
[[1,1,1,2]]
=> 1
[[1,1,2,2]]
=> 1
[[1,2,2,2]]
=> 1
[[2,2,2,2]]
=> 1
[[1,1,1],[2]]
=> 0
[[1,1,2],[2]]
=> 1
[[1,2,2],[2]]
=> 1
[[1,1],[2,2]]
=> 0
[[1,1,1,1,2]]
=> 1
[[1,1,1,2,2]]
=> 1
[[1,1,2,2,2]]
=> 1
[[1,2,2,2,2]]
=> 1
[[2,2,2,2,2]]
=> 1
[[1,1,1,1],[2]]
=> 0
[[1,1,1,2],[2]]
=> 1
[[1,1,2,2],[2]]
=> 1
[[1,2,2,2],[2]]
=> 1
[[1,1,1],[2,2]]
=> 0
[[1,1,2],[2,2]]
=> 1
[[1,1,1,1,1,2]]
=> 1
[[1,1,1,1,2,2]]
=> 1
[[1,1,1,2,2,2]]
=> 1
[[1,1,2,2,2,2]]
=> 1
[[1,2,2,2,2,2]]
=> 1
[[2,2,2,2,2,2]]
=> 1
[[1,1,1,1,1],[2]]
=> 0
[[1,1,1,1,2],[2]]
=> 1
[[1,1,1,2,2],[2]]
=> 1
[[1,1,2,2,2],[2]]
=> 1
[[1,2,2,2,2],[2]]
=> 1
[[1,1,1,1],[2,2]]
=> 0
[[1,1,1,2],[2,2]]
=> 1
[[1,1,2,2],[2,2]]
=> 1
[[1,1,1],[2,2,2]]
=> 0
Description
The segment statistic of a semistandard tableau.
Let ''T'' be a tableau. A ''k''-segment of ''T'' (in the ''i''th row) is defined to be a maximal consecutive sequence of ''k''-boxes in the ith row. Note that the possible ''i''-boxes in the ''i''th row are not considered to be ''i''-segments. Then seg(''T'') is the total number of ''k''-segments in ''T'' as ''k'' varies over all possible values.
Matching statistic: St000845
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Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000845: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000845: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1],[2]]
=> ([],1)
=> 0
[[1,3]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> 0
[[1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> 0
[[1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> 0
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> 0
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> 0
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> 0
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> 0
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> 0
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St000846
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Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1],[2]]
=> ([],1)
=> 0
[[1,3]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> 0
[[1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> 0
[[1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> 0
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> 0
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> 0
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> 0
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> 0
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> 0
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St000272
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Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
Description
The treewidth of a graph.
A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.
Matching statistic: St000536
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Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
Description
The pathwidth of a graph.
Matching statistic: St000537
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
Description
The cutwidth of a graph.
This is the minimum possible width of a linear ordering of its vertices, where the width of an ordering $\sigma$ is the maximum, among all the prefixes of $\sigma$, of the number of edges that have exactly one vertex in a prefix.
Matching statistic: St000778
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
Description
The metric dimension of a graph.
This is the length of the shortest vector of vertices, such that every vertex is uniquely determined by the vector of distances from these vertices.
Matching statistic: St001270
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
Description
The bandwidth of a graph.
The bandwidth of a graph is the smallest number $k$ such that the vertices of the graph can be
ordered as $v_1,\dots,v_n$ with $k \cdot d(v_i,v_j) \geq |i-j|$.
We adopt the convention that the singleton graph has bandwidth $0$, consistent with the bandwith of the complete graph on $n$ vertices having bandwidth $n-1$, but in contrast to any path graph on more than one vertex having bandwidth $1$. The bandwidth of a disconnected graph is the maximum of the bandwidths of the connected components.
Matching statistic: St001271
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
Description
The competition number of a graph.
The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x, v)$ and $(y, v)$ are arcs of $D$. For any graph, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ is the smallest number of such isolated vertices.
Matching statistic: St001277
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> ([],1)
=> 0
Description
The degeneracy of a graph.
The degeneracy of a graph $G$ is the maximum of the minimum degrees of the (vertex induced) subgraphs of $G$.
The following 350 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001352The number of internal nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001644The dimension of a graph. St001792The arboricity of a graph. St001962The proper pathwidth of a graph. St000378The diagonal inversion number of an integer partition. St001580The acyclic chromatic number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001746The coalition number of a graph. St001883The mutual visibility number of a graph. St000259The diameter of a connected graph. St001512The minimum rank of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001029The size of the core of a graph. St001093The detour number of a graph. St001486The number of corners of the ribbon associated with an integer composition. St001494The Alon-Tarsi number of a graph. St001654The monophonic hull number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000822The Hadwiger number of the graph. St001642The Prague dimension of a graph. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001349The number of different graphs obtained from the given graph by removing an edge. St001393The induced matching number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000299The number of nonisomorphic vertex-induced subtrees. St001261The Castelnuovo-Mumford regularity of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001330The hat guessing number of a graph. St000741The Colin de Verdière graph invariant. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001498The normalised height of a Nakayama algebra with magnitude 1. St000006The dinv of a Dyck path. St001488The number of corners of a skew partition. St000100The number of linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000456The monochromatic index of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001531Number of partial orders contained in the poset determined by the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000260The radius of a connected graph. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000914The sum of the values of the Möbius function of a poset. St000736The last entry in the first row of a semistandard tableau. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St000929The constant term of the character polynomial of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000137The Grundy value of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000003The number of standard Young tableaux of the partition. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000053The number of valleys of the Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000075The orbit size of a standard tableau under promotion. St000079The number of alternating sign matrices for a given Dyck path. St000088The row sums of the character table of the symmetric group. St000108The number of partitions contained in the given partition. St000117The number of centered tunnels of a Dyck path. St000148The number of odd parts of a partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000179The product of the hook lengths of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000288The number of ones in a binary word. St000289The decimal representation of a binary word. St000290The major index of a binary word. St000291The number of descents of a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000306The bounce count of a Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000331The number of upper interactions of a Dyck path. St000335The difference of lower and upper interactions. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000391The sum of the positions of the ones in a binary word. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000443The number of long tunnels of a Dyck path. St000475The number of parts equal to 1 in a partition. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000517The Kreweras number of an integer partition. St000529The number of permutations whose descent word is the given binary word. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000543The size of the conjugacy class of a binary word. St000549The number of odd partial sums of an integer partition. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000644The number of graphs with given frequency partition. St000655The length of the minimal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000691The number of changes of a binary word. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000712The number of semistandard Young tableau of given shape, with entries at most 4. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000783The side length of the largest staircase partition fitting into a partition. St000792The Grundy value for the game of ruler on a binary word. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000847The number of standard Young tableaux whose descent set is the binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000922The minimal number such that all substrings of this length are unique. St000935The number of ordered refinements of an integer partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001118The acyclic chromatic index of a graph. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001129The product of the squares of the parts of a partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001249Sum of the odd parts of a partition. St001256Number of simple reflexive modules that are 2-stable reflexive. St001267The length of the Lyndon factorization of the binary word. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001281The normalized isoperimetric number of a graph. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001372The length of a longest cyclic run of ones of a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001389The number of partitions of the same length below the given integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001437The flex of a binary word. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001484The number of singletons of an integer partition. St001485The modular major index of a binary word. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001525The number of symmetric hooks on the diagonal of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001571The Cartan determinant of the integer partition. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001838The number of nonempty primitive factors of a binary word. St001884The number of borders of a binary word. St001910The height of the middle non-run of a Dyck path. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001984A recursive count of subwords of the form 01, 10 and 11. St000454The largest eigenvalue of a graph if it is integral. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000934The 2-degree of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001645The pebbling number of a connected graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000928The sum of the coefficients of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
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